CEMC flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

For ax²+bx+c=0 with a≠0, the roots are x=(−b±√(b²−4ac))/(2a). The discriminant b²−4ac determines whether roots are real and distinct, repeated, or complex.

Remainder theorem

When a polynomial P(x) is divided by (x−r), the remainder equals P(r). Consequently, (x−r) is a factor exactly when P(r)=0.

Function composition domain

The domain of f∘g consists of all x in the domain of g for which g(x) lies in the domain of f. Composition order generally changes the result.

Inverse function condition

A function has an inverse on its domain exactly when it is one-to-one there, which holds if it is strictly increasing or strictly decreasing throughout that domain.

AM-GM inequality

For positive numbers, the arithmetic mean is at least the geometric mean: (a+b)/2 ≥ √(ab), with equality only when a=b. It generalizes to n terms.

Law of cosines

In a triangle with sides a, b, c and angle C opposite side c: c²=a²+b²−2ab cos C. It extends the Pythagorean theorem to non-right triangles.

Heron's formula

The area of a triangle with sides a, b, c equals √(s(s−a)(s−b)(s−c)), where s=(a+b+c)/2 is the semiperimeter.

Sector area in radians

For a circle of radius r and central angle θ measured in radians, arc length is rθ and sector area is ½r²θ.

Double angle formulas

sin 2θ=2 sin θ cos θ, and cos 2θ=cos²θ−sin²θ, which also equals 1−2sin²θ and 2cos²θ−1.

Regular polygon interior angle

The interior angles of a regular n-gon each measure (n−2)·180°/n degrees, since the total angle sum of any n-gon is (n−2)·180°.

Sphere volume and surface

A sphere of radius r has volume 4πr³/3 and surface area 4πr². These formulas also underlie related cone and cylinder comparisons.

Euclidean algorithm

Repeatedly replace the larger number by its remainder upon division by the smaller; the last nonzero remainder is gcd(a,b). It terminates because remainders strictly decrease.

gcd-lcm product identity

For positive integers a and b, gcd(a,b)·lcm(a,b)=ab. This lets you compute one quantity from the other once the gcd is known.

Divisibility by 3 and 9

An integer is divisible by 3, respectively 9, exactly when the sum of its decimal digits is divisible by 3, respectively 9.

Modular arithmetic rules

If a≡b (mod m) and c≡d (mod m), then a+c≡b+d, a−c≡b−d, and ac≡bd (mod m). Division requires the divisor to be invertible modulo m.

Fermat's little theorem

If p is prime and a is not divisible by p, then a^(p−1)≡1 (mod p). Equivalently, aᵖ≡a (mod p) for every integer a.

Eulerian circuit criterion

A connected graph has a closed trail using every edge exactly once precisely when every vertex has even degree. Exactly two odd vertices gives an Eulerian path instead.

Arithmetic series sum

For the sequence a, a+d, a+2d, …, the sum of the first n terms is n(2a+(n−1)d)/2, equivalently n times the average of first and last terms.

Infinite geometric series

The sum a+ar+ar²+… equals a/(1−r) when |r|<1, and diverges when |r|≥1 unless a=0.

Power sum formulas

Σk from 1 to n equals n(n+1)/2; Σk² equals n(n+1)(2n+1)/6; Σk³ equals [n(n+1)/2]².

Monotone convergence principle

A sequence that is monotonic (always nondecreasing or nonincreasing) and bounded always converges. Boundedness alone or monotonicity alone does not guarantee convergence.

Harmonic series divergence

The series 1+1/2+1/3+… diverges even though its terms tend to zero. Vanishing terms are necessary but not sufficient for convergence.

Ratio test

For a series with positive terms, compute L=lim|aₙ₊₁/aₙ|. The series converges if L<1, diverges if L>1, and the test is inconclusive when L=1.

Combinations formula

The number of r-element subsets of an n-element set is C(n,r)=n!/(r!(n−r)!). It satisfies the symmetry C(n,r)=C(n,n−r).

Pigeonhole principle

If more than n objects are placed into n containers, at least one container holds two or more objects. Generalized forms count ceilings of object-to-container ratios.

Inclusion-exclusion principle

For two sets, |A∪B|=|A|+|B|−|A∩B|. For three, subtract pairwise intersections and add back the triple intersection to avoid double counting.

Conditional probability formula

For events with P(B)>0, P(A|B)=P(A∩B)/P(B). Independence means P(A|B)=P(A), equivalently P(A∩B)=P(A)P(B).

Binomial probability

In n independent trials each with success probability p, the probability of exactly k successes is C(n,k)pᵏ(1−p)^(n−k).

Proof by induction

Establish the statement for a base case, assume it holds for n=k, then use that assumption to prove it for n=k+1, concluding it holds for all integers beyond the base.

Proof by contradiction

To prove a claim, assume its negation and derive a logical impossibility or a conflict with a known fact; the negation is therefore false and the claim true.

Invariant principle

Identify a quantity unchanged by every allowed operation. If a target state has a different invariant value, it cannot be reached from the starting state.

Extremal principle

Choose a maximal or minimal element among all configurations satisfying the conditions, then exploit that extremality to force a contradiction or structural conclusion.